When transmitters and receivers are synchronized with one another within a transmission system, the clock-pulse and carrier signal are matched respectively with regard to phase position and frequency at the transmitter end and the receiver end. The clock-pulse synchronization to be considered in the present description requires a clock-pulse recovery in the receiver, which can be realized with or without feedback.
In the case of the clock-pulse recovery with feedback, the clock-pulse phase and frequency are estimated on the basis of the received signal, and a frequency oscillator for phase-synchronous and frequency-synchronous sampling of the received signal is post-tuned at the correct inter-symbol interference-free decision timing points.
By contrast, in the case of a feedback-free clock-pulse recovery, the clock-pulse phase and frequency are estimated on the basis of the received signal sampled with a fixed sampling frequency, and the correct symbol value of the received signal at the respective decision timing point is determined via an interpolator from the sampled values, which are adjacent to the respective inter-symbol interference-free decision timing points.
For the feedback-free clock-pulse recovery with a fixed clock-frequency, which is known to the receiver, a method for pulse amplitude modulated (PAM), quadrature-phase-modulated (QPSK) and π/4 quadrature-phase-modulated (π/4-QPSK) signals based on the maximum-likelihood estimation is detailed in K. Schmidt: “Digital clock-pulse recovery for bandwidth-efficient mobile telephone systems”, 1994, ISBN 3-18-14 7510-6.
In this context, the maximum-likelihood estimation is based upon a maximisation of the likelihood function, which, via an inverse exponential function, minimizes the modulus error squared between the measured, noise-contaminated received signal and a model, ideal, noise-free transmission signal containing the desired timing offset over an observation period. The desired timing offset is obtained, when the model transmission signal approximates the measured receiver signal with minimal modulus error squared.
As shown in detail in the K. Schmidt article and below, the likelihood function is obtained from the received signal, convoluted with the impulse response of a signal-matched pre-filter, which is subjected, after pre-filtering, to a nonlinear function and then averaged over a limited number of symbols. As also shown by the K. Schmidt article, the nonlinear function can be approximated by a modulus squaring. If the timing offset is determined in the time domain, the desired timing offset is obtained from a maximum detection of the pre-filtered, modulus-squared and averaged received signal, which corresponds to the maximum-likelihood function.
The disadvantage of an inaccurate or ambiguous maximum detection in the time domain resulting from inadequate removal of interference in the useful signal can be avoided by observation in the frequency domain. A determination of the timing offset in the frequency domain, exploits the fact that the pre-filtered, modulus-squared received signal averaged over a limited number of symbols provides a basic periodicity over the symbol length and respectively a maximum at multiples of the symbol length. The timing offset can therefore be determined according to a discrete Fourier transformation of the pre-filtered, modulus-square received signal averaged over a given number of symbols from the phase of the spectral line at the basic spectral frequency determined by the symbol frequency.
The frequency-domain-orientated determination of the timing offset outlined above, which presupposes a spectral line corresponding to a periodized received signal, fails in the case of an offset quadrature-phase-shift keying (OQPSK) modulated signal. As a result of the modulus squaring, the in-phase and quadrature components of the pre-filtered offset-QPSK-received signal, phase-offset by one symbol length, cancel each other out with regard to their alternating component and lead to a time signal, which provides substantially only a direct component and therefore no spectral lines containing the timing offset.